Reproducible Detailed-Balance Benchmarking of Single-Junction Photovoltaics Under ASTM G173 AM1.5G and ASTM E490 AM0 Spectra

1. Introduction

The detailed-balance limit establishes the conversion ceiling of an ideal photovoltaic junction by requiring consistency between absorption and radiative emission [1]. Later analyses refined the spectral inputs, loss partitions, and material interpretations [2,3]. The limit remains a reference for judging how far a measured device is from its radiative potential, but a reported numerical value is not unique unless the incident spectrum, cell temperature, emission solid angle, rear optical boundary, integration window, and numerical resolution are all stated.

This qualification matters when terrestrial AM1.5G and extraterrestrial AM0 conditions are compared. Their total irradiances, photon distributions, and atmospheric absorption structures differ, so equal bandgaps need not yield equal current, voltage, or optimum efficiency. Recent work continues to reassess the interpretation of the Shockley–Queisser limit [4]. Here, established detailed-balance equations are implemented in a common workflow to examine spectral-window selection, the optical emission boundary, and numerical sensitivity. The contribution is reproducibility and transparent comparison; no new conversion mechanism or fundamental limit is proposed.

Thermodynamic terminology is used here in its established photovoltaic sense. The carrier chemical potential qV is the work-bearing quantity of an illuminated junction, and the entropy cost of emission and angular dilution enters through the radiative voltage balance [5,6]. No information-theoretic entropy is subtracted from the incident energy flux.

The contributions of this study are:

a common, reproducible calculation using the ASTM G173 AM1.5G table and the full ASTM E490 AM0 table;

explicit one- and two-emitting-side boundary conditions, with a published AM1.5G benchmark;

a closed-form Lambert-W maximum-power solution checked by a separate numerical optimization of the same diode model;

bandgap-grid convergence, spectral-resampling sensitivity, an algebraic loss-accounting decomposition, and a deterministic parameter envelope.

2. Materials and Methods

2.1. Reference Spectra and Data Conditioning

The terrestrial input is the Global Tilt column of ASTM G173-03(2020), which is based on SMARTS calculations for a 37° tilted surface [7,8]. It contains 2002 points from 280 to 4000 nm. The extraterrestrial input is the full ASTM E490-00a(2019) table from 119.5 nm to 1 mm [9]. Native sampling is retained for the principal integrations. Linear interpolation is used only to insert an absorption cutoff between adjacent tabulated wavelengths and for the declared uniform-grid sensitivity cases. The processed input files are taken from the public reference-spectrum tables hosted by the National Laboratory of the Rockies (https://www.nlr.gov/grid/solar-resource/spectra-am1.5 and https://www.nlr.gov/grid/solar-resource/spectra-astm-e490). These historical reference spectra are used as fixed modeling inputs, not as measurements of present-day solar irradiance.

Table 1. Reference spectral inputs used in the calculations.

Spectrum	Integration window	Points	Integrated irradiance (W m−2)
ASTM G173 AM1.5G	280–4000 nm	2002	1000.371
ASTM E490 AM0	119.5 nm–1 mm	1697	1366.091

Table 1. Reference spectral inputs used in the calculations.

Spectrum	Integration window	Points	Integrated irradiance (W m−2)
ASTM G173 AM1.5G	280–4000 nm	2002	1000.371
ASTM E490 AM0	119.5 nm–1 mm	1697	1366.091

Figure 1. ASTM G173 AM1.5G and ASTM E490 AM0 spectral irradiance and photon flux. The display is limited to 280–2500 nm for legibility; the AM0 calculation uses the full E490 table.

2.2. Photon Flux and Detailed Balance

The tabulated spectral irradiance Iλ is in W m⁻² nm⁻¹. Photon energy is evaluated using wavelength λm in meters; the resulting photon flux φλ remains per nanometer:

$$E_{\gamma }={\displaystyle \frac{\mathit{hc}}{{\lambda }_{\mathrm{m}}}}, {\varphi }_{\lambda }={\displaystyle \frac{I_{\lambda }}{E_{\gamma }}}$$

(1)

The model uses unit absorptance for photon energies at or above Eg and zero absorptance below it, with one collected electron per absorbed photon. The cutoff λg is expressed in nanometers for the following integral. Spectral flux is taken as zero outside the supplied table; this is an explicit finite-window convention, not a claim that physical solar radiation is zero there:

$$J_{\mathrm{sc}}\left(E_g\right)=q{\int }_0^{{\lambda }_g}{\varphi }_{\lambda }\left(\lambda \right)\hspace{0.17em}\mathrm{d}\lambda$$

(2)

Absorption and emission use the same ideal step-function spectral boundary [10,11]. Unlike the finite tabulated solar input, the thermal-emission integral extends to infinite photon energy. The radiative dark saturation current is evaluated using dimensionless photon energy x = E/(kBTc):

$$J_{0,\mathrm{rad}}=\mathit{sq}{\displaystyle \frac{2\pi {\left(k_{\mathrm{B}}{T}_c\right)}^3}{h^3{c}^2}}{\int }_{x_g}^{\mathrm{\infty }}{\displaystyle \frac{x^2}{\exp \left(x\right)-1}}\hspace{0.17em}\mathrm{d}x, x_g={\displaystyle \frac{E_g}{k_{\mathrm{B}}{T}_c}}$$

(3)

where s = 1 represents one emitting hemisphere with an ideal rear reflector and s = 2 represents equal front and rear emission. The illuminated ideal-diode relation and radiative open-circuit voltage are

$$J\left(V\right)=J_{\mathrm{sc}}-J_0\left[\exp \left({\displaystyle \frac{\mathit{qV}}{k_{\mathrm{B}}{T}_c}}\right)-1\right]$$

(4)

$$V_{\mathrm{oc}}={\displaystyle \frac{k_{\mathrm{B}}{T}_c}{q}}\ln \left(1+{\displaystyle \frac{J_{\mathrm{sc}}}{J_0}}\right)$$

(5)

External luminescence and optical extraction are therefore voltage-relevant loss channels rather than optional corrections [12,13].

2.3. Closed-Form Maximum-Power Point

Differentiating P(V) = VJ(V) gives an analytic maximum-power solution. With Vt = kBTc/q and z = Vmpp/Vt,

$$z=W_0\left[\exp \left(1\right)\left(1+{\displaystyle \frac{J_{\mathrm{sc}}}{J_0}}\right)\right]-1, V_{\mathrm{mpp}}=z{V}_t$$

(6)

where W is the principal Lambert-W function [14]. The maximum power density, fill factor, and efficiency are Pmpp = VmppJ(Vmpp), FF = Pmpp/(JscVoc), and η = Pmpp/Pin, respectively. A separate bounded scalar maximization of VJ(V) is retained as a numerical consistency check. The elementary charge q is distinguished from Euler’s number exp(1). Energies are converted from eV to joules in Equation (3). The calculation uses SI constants h = 6.62607015 × 10⁻³⁴ J s, c = 299792458 m s⁻¹, q = 1.602176634 × 10⁻¹⁹ C, and kB = 1.380649 × 10⁻²³ J K⁻¹.

2.4. Aggregate Voltage-Deficit Sensitivity

A dimensionless voltage-retention ratio rV is used only to examine the effect of an imposed aggregate voltage deficit relative to the radiative open-circuit voltage:

V_oc,eff = r_VV_oc,rad,  0 < r_V ≤ 1

(7)

For each value, J0,eff = Jsc/{exp[Voc,eff/Vt] - 1} is inserted into Equation (4), and the maximum-power point is recalculated. This ratio is not a material constant or a new device law. It does not alter absorptance, generate carriers, or represent spectral conversion; it only imposes a specified voltage deficit for sensitivity analysis.

2.5. Temperature-Dependent Silicon Case

For an ideal silicon illustration, the bandgap is varied with the Varshni relation [15] using the coefficients employed in the extreme-temperature photovoltaic study of Libra et al. [16]: Temperature T in Equation (8) is in kelvin; the coefficients are 1.16 eV, 7.02 × 10⁻⁴ eV K⁻¹, and 1108 K.

$$E_g\left(T\right)=1.16\hspace{0.17em}\mathrm{eV}-{\displaystyle \frac{\left(7.02\times {10}^{-4}\hspace{0.17em}\mathrm{eV}\hspace{0.17em}{\mathrm{K}}^{-1}\right)T^2}{T+1108\hspace{0.17em}\mathrm{K}}}$$

(8)

At every temperature from −100 to 100 °C, Eg(T), Jsc, J0,rad, Voc, and Pmpp are recalculated under fixed AM1.5G illumination. This isolates radiative and bandgap effects. It does not model measured series resistance, thermal expansion, contact degradation, or a specific device package; those mechanisms explain why experimental temperature coefficients can differ [17,18].

2.6. Numerical Checks and Sensitivity Design

Four checks address different questions. Integrated spectral powers are compared with the reference-table normalizations. The two-emitting-side terrestrial calculation is repeated at 298.15 K to compare its rounded maximum with Rühle’s standard-test-condition value of 33.16% at approximately 1.34 eV [3]. The comparison isolates temperature within the present implementation; it is not a complete reproduction of the reference implementation. Equation (6) is compared with bounded numerical maximization over 54 bandgaps; both calculations share the same diode currents. Finally, a four-term loss partition is checked for arithmetic consistency. Because its residual loss is defined by subtraction, power closure is an algebraic identity and cannot independently validate the physical model.

The sensitivity analysis uses explicit parameter ranges rather than unsupported probability distributions [19]. Bandgap steps from 20 to 0.2 meV and uniform spectral steps from 5 to 0.25 nm quantify numerical effects. Parametric ranges are 298-302 K and a correlated irradiance scale of 0.98-1.02. Optical model form is assessed separately by comparing one and two emitting sides. Voltage retention from 1.00 to 0.85 is reported as an imposed performance scenario, not as an uncertainty interval. A deterministic corner envelope combines temperature, irradiance scale, and spectral grid at fixed one-sided emission.

The analysis was implemented in Python using NumPy, SciPy, Matplotlib, and pandas [24,25,26,27]. Processed spectra, code, machine-readable outputs, and convergence tables accompany the manuscript. Native-grid solar integrations use the trapezoidal rule with a linearly interpolated photon-flux cutoff. The blackbody integral is evaluated as an exponential series with 12 terms. Bandgaps from 0.6 to 2.4 eV are scanned at 0.5 meV spacing for the central results; the reported optimum is a grid maximum. AI assistance in code development and manuscript preparation is disclosed below.

3. Results

3.1. Spectral Verification and Radiative Optima

The AM1.5G table integrates to 1000.371 W m⁻², only 0.037% above the nominal 1000 W m⁻² reference. The complete E490 table integrates to 1366.091 W m⁻², within 0.001% of the 1366.1 W m⁻² reference. These checks confirm that the principal calculations use the intended spectral windows and denominators.

Figure 2. (Left) AM1.5G efficiency for one and two emitting sides, with the Rühle [3] benchmark. (Right) One-side AM1.5G and AM0 efficiency versus bandgap. Curves use 300 K; the literature marker uses 298.15 K (see Section S9).

Table 2. Calculated radiative-limit optima at 300 K.

Spectrum / boundary	Eg (eV)	Jsc (mA cm−2)	Voc (V)	FF	η (%)	Pmpp (W m−2)
AM1.5G, one side	1.3365	35.164	1.0785	0.8888	33.693	337.06
AM1.5G, two sides	1.3365	35.164	1.0605	0.8873	33.080	330.92
AM0, one side	1.2450	46.936	0.9980	0.8820	30.243	413.15
AM0, two sides	1.2610	46.170	0.9950	0.8817	29.651	405.06

Table 2. Calculated radiative-limit optima at 300 K.

Spectrum / boundary	Eg (eV)	Jsc (mA cm−2)	Voc (V)	FF	η (%)	Pmpp (W m−2)
AM1.5G, one side	1.3365	35.164	1.0785	0.8888	33.693	337.06
AM1.5G, two sides	1.3365	35.164	1.0605	0.8873	33.080	330.92
AM0, one side	1.2450	46.936	0.9980	0.8820	30.243	413.15
AM0, two sides	1.2610	46.170	0.9950	0.8817	29.651	405.06

Rühle reports a maximum of 33.16% at approximately 1.34 eV for standard test conditions at 298.15 K [3]. Comparing this directly with the present 300 K result mixes temperatures. Holding the spectrum, integration procedure, bandgap grid, and two-emitting-side boundary fixed, the present maximum rises from 33.079635% at 300 K to 33.163482% at 298.15 K, both at 1.3365 eV. The 0.083847-percentage-point temperature shift accounts for the magnitude of the earlier 0.080-percentage-point difference. At 298.15 K the computed value is 0.003482 percentage points above 33.16% and agrees at the published two-decimal precision. This is a temperature-controlled consistency comparison; it does not demonstrate identical optical or spectral-processing assumptions in the two implementations. Supplementary Section S9 provides the reproducible calculation.

Using the complete E490 denominator gives a one-side AM0 optimum of 30.243% at 1.2450 eV and 413.15 W m⁻². Its percentage efficiency is lower than the AM1.5G limit, but its absolute output is higher because the extraterrestrial incident power is 36.6% larger. Restricting the E490 denominator to 280–4000 nm would favorably bias the reported AM0 percentage; the present result avoids that truncation.

3.2. Numerical Validation and Convergence

Figure 3. Bandgap-grid convergence, spectral-resampling sensitivity, and analytic Lambert-W versus numerical maximum-power validation for AM1.5G.

Table 3. Numerical checks and their interpretation; these are not experimental validation criteria.

Check	Result	Interpretation
AM1.5G integral	1000.371 W m−2	0.037% from nominal 1000 W m−2
Full E490 integral	1366.091 W m−2	<0.001% from 1366.1 W m−2
Rühle comparison at 298.15 K	33.1635% vs 33.16%	Same rounded maximum; full conventions not independently matched
Lambert-W MPP	max |ΔV| = 8.64e-09 V	max relative ΔP = 1.02e-15
Energy closure	max residual = 0.0e+00 W m−2	Identity by construction; bookkeeping only
Fine Eg grids	η span = 0.0007 pp	Steps ≤1 meV

Table 3. Numerical checks and their interpretation; these are not experimental validation criteria.

Check	Result	Interpretation
AM1.5G integral	1000.371 W m−2	0.037% from nominal 1000 W m−2
Full E490 integral	1366.091 W m−2	<0.001% from 1366.1 W m−2
Rühle comparison at 298.15 K	33.1635% vs 33.16%	Same rounded maximum; full conventions not independently matched
Lambert-W MPP	max |ΔV| = 8.64e-09 V	max relative ΔP = 1.02e-15
Energy closure	max residual = 0.0e+00 W m−2	Identity by construction; bookkeeping only
Fine Eg grids	η span = 0.0007 pp	Steps ≤1 meV

The analytic and numerical maximum-power calculations are effectively identical: the largest voltage difference is 8.64 nV and the largest relative power difference is 1.02×10⁻¹⁵. Fine bandgap meshes change the reported maximum by only 0.0007 percentage points. Uniform spectral grids of 0.25–5 nm span 0.079 percentage points because coarse interpolation smooths narrow atmospheric features and slightly changes integrated power. The native grid is therefore retained for the central result.

3.3. Deterministic Sensitivity and Model-Form Effects

Figure 4. Deterministic AM1.5G sensitivity ranges relative to the one-emitting-side 300 K baseline. The intervals are not statistical confidence intervals.

Table 4. AM1.5G sensitivity ranges at re-optimized bandgap.

Source	Range	Minimum η (%)	Maximum η (%)	Span (pp)	Classification
Cell temperature	298–302 K	33.607	33.780	0.173	Parametric
Irradiance scale	0.98–1.02	33.675	33.711	0.035	Parametric
Uniform spectral grid	0.25–5 nm	33.672	33.751	0.079	Numerical
Emission boundary	1–2 sides	33.080	33.693	0.614	Model form
Voltage retention	1.00–0.95	31.848	33.693	1.846	Performance scenario

Table 4. AM1.5G sensitivity ranges at re-optimized bandgap.

Source	Range	Minimum η (%)	Maximum η (%)	Span (pp)	Classification
Cell temperature	298–302 K	33.607	33.780	0.173	Parametric
Irradiance scale	0.98–1.02	33.675	33.711	0.035	Parametric
Uniform spectral grid	0.25–5 nm	33.672	33.751	0.079	Numerical
Emission boundary	1–2 sides	33.080	33.693	0.614	Model form
Voltage retention	1.00–0.95	31.848	33.693	1.846	Performance scenario

At fixed one-sided emission, the deterministic corner envelope combining temperature, irradiance scale, and spectral grid is 33.568-33.797% (span 0.229 percentage points). This range is not a confidence interval because no measurement covariance or probability model was supplied. Optical boundary and voltage retention are reported separately because they represent a device configuration and an imposed performance scenario. Reducing rV from 1.00 to 0.85 lowers the optimized efficiency to 28.165%.

3.4. Temperature-Dependent Silicon Illustration

Figure 5. Ideal silicon radiative-limit efficiency and open-circuit voltage under AM1.5G from −100 to 100 °C using the Varshni bandgap relation.

The ideal silicon case decreases from approximately 39.9% at −100 °C to 29.0% at 100 °C. Near 25 °C, the model gives an absolute efficiency slope of about −0.062 percentage points K⁻¹ and a Voc slope near −1.15 mV K⁻¹. These values describe only radiative recombination plus Eg(T). The agreement in direction and order of magnitude with measured temperature trends [17,18] is supportive, not a device-level validation.

3.5. Energy-Loss Closure

Figure 6. Incident-power partition at each spectrum’s one-emitting-side radiative optimum. Electrical output, voltage and fill-factor loss, thermalization, and sub-bandgap transmission sum to 100%.

At each selected optimum, incident power is partitioned into sub-bandgap transmission, above-bandgap thermalization, electrical output, and a residual voltage/fill-factor term. Let Pabs be the integrated above-bandgap irradiance and Pgap = Eg Jsc/q with Eg in joules. The terms are Pin − Pabs, Pabs − Pgap, Pmpp, and Pgap − Pmpp, respectively. Their sum equals Pin by construction. The residual term aggregates conversion losses; it is not a separately calculated radiative heat flux or an independent entropy balance.

4. Discussion

4.1. What the Numerical Checks Establish

The numerical checks support the implementation of the ideal-diode maximum-power calculation and show the sensitivity of its result to specified discretizations. The temperature-controlled literature comparison reproduces the reported maximum to two decimal places. Agreement of a rounded maximum alone does not independently establish all physical conventions or validate the full efficiency curve. Algebraic loss closure checks bookkeeping only. These results do not validate a measured device, establish a new physical law, or replace an independent implementation of the full model. Material absorption edges, nonradiative recombination, parasitic optical loss, series resistance, shunting, and imperfect carrier collection reduce practical performance [11,12,13,20].

The 0.614-percentage-point difference between one- and two-side terrestrial cases is larger than the combined numerical and modest parametric envelope. Consequently, an efficiency value quoted without its rear optical boundary is under-specified. This controlled comparison demonstrates a boundary-condition sensitivity within the present model; it does not by itself reconcile numerical differences across publications. Optical étendue and luminescence extraction are part of the thermodynamic boundary, not formatting details [10,21].

4.2. Interpretation of AM1.5G and AM0

AM0 has more incident power and yields more electrical watts per square meter at its optimum, but the additional extraterrestrial spectral power does not increase percentage efficiency in the present one-sun, single-junction model. The optimum bandgap shifts downward to about 1.245 eV because the full E490 photon distribution changes the balance among current, thermalization, and sub-bandgap transmission. This conclusion is specific to one junction, unity step absorptance, fixed 300 K cell temperature, and no concentration. It does not include orbital distance, charged-particle damage, coverglass transmission, or thermal-control design.

4.3. Interpretation of the Voltage-Retention Ratio

The bounded ratio rV provides a compact sensitivity test for aggregate voltage loss. Because rV ≤ 1 and the diode saturation current is recalculated consistently, the scenario cannot exceed the radiative reference. The ratio must not be interpreted as a unique recombination mechanism or universal material parameter: similar voltage losses can arise from bulk defects, interfaces, contact selectivity, photon escape, or several mechanisms together. Device-specific analysis should instead connect external radiative efficiency, electroluminescence, and voltage loss through reciprocity [11,12,13].

4.4. Limitations and Future Work

The model assumes step-function absorptance, unity carrier collection, a radiative ideality factor of one, and spatially uniform temperature. No measured current-voltage curve or material-specific optical constants are fitted. The deterministic sensitivity ranges therefore do not replace measurement-based uncertainty analysis. These assumptions must accompany the reported limits so the calculations are not mistaken for certified device predictions.

The next validation step is to reproduce the calculation with an independent detailed-balance code and to compare material-specific absorptance and external luminescence data. Extensions to hot-carrier extraction and intermediate-band transitions require modified carrier and photon balances rather than an efficiency multiplier [22,23]. Spectral splitting, tandem junctions, angular restriction, and temperature-dependent optical constants are also natural extensions.

5. Conclusions

The supplied implementation compares ideal single-junction conversion under ASTM G173 AM1.5G and full-spectrum ASTM E490 AM0 illumination. At 300 K, the one-emitting-side AM1.5G maximum is 33.693% at 1.3365 eV, compared with 33.080% for two emitting sides. The full-spectrum, one-emitting-side AM0 maximum is 30.243% at 1.2450 eV and 413.15 W m⁻². Analytic and numerical optimizations of the same diode model agree near floating-point precision over the tested cases; fine bandgap grids give a terrestrial efficiency span below 0.001 percentage points. The deterministic AM1.5G sensitivity envelope is 33.568–33.797%, while changing the emission boundary produces a 0.614-percentage-point shift. These results illustrate the importance of reporting spectral coverage, temperature, optical boundaries, and numerical conventions. A separate two-emitting-side calculation at 298.15 K gives 33.1635%, consistent with the published 33.16% after rounding. Independent physical validation and complete reproduction of the reference implementation remain outside the evidence presented here.